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Pasting polynomials together Sarah C. Koch University of - - PowerPoint PPT Presentation
Pasting polynomials together Sarah C. Koch University of - - PowerPoint PPT Presentation
Pasting polynomials together Sarah C. Koch University of Michigan The Basilica San Marco Cathedral Venice, Italy The Rabbit p ( z ) = z 2 +( 0 . 1226+0 . 7449 i ) The Corabbit p ( z ) = z 2 +( 0 . 1226 0 . 7449 i ) A
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The Rabbit
− p(z) = z2+(−0.1226+0.7449i)
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The Corabbit
− p(z) = z2+(−0.1226−0.7449i)
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− p(z) = z2 + i
A dendrite
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Kokopelli
p(z) = z2 0.156 + 1.302i
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Cokokopelli
p(z) = z2 0.156 1.302i
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A Cantor set
p(z) = z2 + (0.5 + i)
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z2
z2 − 1 basilica
− z2 + (−0.1 + 0.75i) rabbit − z2 + (−0.1 + 0.75i) corabbit
− z2 − 0.156 + 1.302i kokopelli
− z2 + i dendrite
Keeping track of shapes
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c plane
Parameter space: coloring scheme?
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z2
z2 − 1 basilica
− z2 + (−0.1 + 0.75i) rabbit − z2 + (−0.1 + 0.75i) corabbit
− z2 − 0.156 + 1.302i kokopelli
− z2 + i dendrite
The Mandelbrot Set
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the rabbit the basilica
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The mating of the basilica and the rabbit
F(z) = 2z2 + 1 − √ 3 2z2 − 2 C
⇥n
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hmmm... let’s see that again.
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Which quadratic polynomials can be mated?
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- Theorem. (Tan Lei, Rees, Shishikura) Let p : z 7! z2 +c1
and q : z 7! z2 + c2 be postcritically finite. Then p and q can be mated if and only if c1 and c2 do not belong to conjugate limbs of the Mandelbrot Set.
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A shared mating
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Arnaud Cheritat polynomial matings: https://www.math.univ- toulouse.fr/~cheritat/MatMovies/
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Fractal Stream Dynamics Explorer Mandel Software: complex analysis, topology, differential geometry, algebraic topology
Dynamics in one complex variable by John Milnor An introduction to chaotic dynamical systems by Robert Devaney
Books: Classes:
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